MinT - Best Known (112−39, 112, s)-Nets in Base 8

Best Known (112−39, 112, s)-Nets in Base 8

(112−39, 112, 368)-Net over F₈ — Constructive and digital

Digital (73, 112, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (53, 92, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 46, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(112−39, 112, 576)-Net in Base 8 — Constructive

(73, 112, 576)-net in base 8, using

trace code for nets [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(112−39, 112, 1004)-Net over F₈ — Digital

Digital (73, 112, 1004)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−39, 112, 213816)-Net in Base 8 — Upper bound on s

There is no (73, 112, 213817)-net in base 8, because

1 times m-reduction [i] would yield (73, 111, 213817)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17499 377470 745312 306991 213053 571921 956949 858520 648406 620090 432822 639817 727484 385730 417821 696779 597640 > 8¹¹¹ [i]